Problem: Factor the following expression: $-8$ $x^2$ $-17$ $x+$ $21$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(21)} &=& -168 \\ {a} + {b} &=& & & {-17} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-168$ and add them together. Remember, since $-168$ is negative, one of the factors must be negative. The factors that add up to ${-17}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${-24}$ $ \begin{eqnarray} {ab} &=& ({7})({-24}) &=& -168 \\ {a} + {b} &=& {7} + {-24} &=& -17 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-8}x^2 +{7}x {-24}x +{21} $ Group the terms so that there is a common factor in each group: $ ({-8}x^2 +{7}x) + ({-24}x +{21}) $ Factor out the common factors: $ x(-8x + 7) + 3(-8x + 7) $ Notice how $(-8x + 7)$ has become a common factor. Factor this out to find the answer. $(-8x + 7)(x + 3)$